Valence, energy, major/minor and tempo

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In terms of energy/valence the discover weeklies are distributed fairly similar. Some deviations are for instance week 4 and week 6, that seems to have its energy more concentrated in the center. Week 6 also seems to be more optimistic in general. All playlists consist of mostly major songs which, honestly, was to my surprise. Something interesting in weekly 2 is that the songs with highest valence and highest energy are songs in a minor key. Also week 7 shows something remarkable. The happiest most energetic songs here seem to be in the minor keys whilst the songs with lower energy and lower valence seem to be in a major key.

Change of energy in songs in the discover weekly playlists

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In this violin graph you can clearly see that Discover Weekly 4 and 6 have their energy more concentrated whilst the others have it more evenly distributed.

Keys

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It’s interesting to see how my discover weekly playlists seem to start with more peaks regarding the keys they are in. In the third week the peaks have sort of disappeared, then week 4 is mostly in B minor and G major. After this it seems to be relatively evenly distributed for the remaining weeks. D minor and D# minor appear to occur the least amount of times.

Tempo

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Some thing I like to be seeing is that the third week’s discover weekly playlists seem to have an almost gaussian distribution around 125 bpm. This is nice because it coincides with the paper by Moelants and Dirk about prefered tempo.

However, week 6 and 7 show something quite different, being much slower and centered around 100 bpm, which was the idea of preferred tempo before this paper.

Loudness

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In terms of loudness it remains fairly constant across the weeks with one clear outlier being week 3 having a peak at around -7.5 Db.

Chromagraphs for two versions of Paranoid android

Trying to find a dynamic warp path for these two versions is quite hard

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Due to the completely different timings in the song and generally different approach to the song, they are completely different songs according to Dynamic Time Warping. Let’s take a look at another example.

“Hey Joe”, both covers, one very popular, one less popular (but really should be)

DTW path is even less observable now

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Even though when you listen to both songs you can immediately link them (probably due to the lyrics), they are nothing alike due to DTW.

Still, when I listen to these two covers I feel that they are more similar than the two “Paranoid Android” versions which does contain some clearer warping paths (upper right area)

From this we can conclude that finding similarity between songs is not as easy as you may think and that just knowing what the pitches are will not be enough. Apparently some cover songs that are completely different in instruments can be more similar than two covers that share the instruments and lyrics but are performed quite differently. But of course, we haven’t accounted for timbre features yet.

(Side note if you don’t know the Roy Buchanan version and you like the Jimi Hendrix version really give Roy’s version a listen)

Miss Alissa by Eagles of Death metal and Heather by Billy Cobham

So I took a look in terms of valence and energy for two songs that should be polar opposites according to valence and energy and boy is that right.

In the lower left corner we have, with a valence of 0.041, an energy rating of 0.0079 and a length of 8 minutes and 39 seconds, the smooth and shockingly calming “Heather” by Billy Cobham.

In the opposing upper right corner we have, with a valence of 0.877, energy rating of 0.992 and a length of 2 minutes and 38 seconds, you can’t stop dancing to this one, “Miss Alissa” by Eagles of Death Metal.

I analysed differences in chromagrams, cepstograms and self similarity matrices.

Chroma comparisson

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Miss Alissa seems to be all over the place in terms of chroma, whilst Heather is more organized and stays close to F for the whole duration of the song. Of course Heather is a lot longer so it leaves a lot more room for organization and build up.

MFCC comparisson

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Miss Alissa seems to mostly concentrated in the first cepstral coefficients whilst Heather is concentrated more in the second and third.

First cepstral coefficient referring to loudness seems appropriate because Miss Alissa is quite loud and Heather is quite quiet.

Around 220 seconds the saxophone joins the song Heather which is observable in the cepstogram due to the distribution of magnitude getting more spread out.

Self similarity matrices Timbre

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The self similarity matrices in terms of timbre show that Miss Alissa is fairly constant in terms of timbre whilst Heather shows a big change around 220 seconds which is because of the saxophone taking the lead.

Self Similarity matrices Pitch

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Now it’s funny to see that the chromagram of Miss Alissa is complete chaos but the self similarity matrix is fairly constant and looks like one block implying that it might be quite chaotic but the chaos is very constant, which is actually a pretty good dexcription for the song.

On the other hand there is Heather that has a very organized chromagram that looks fairly constant but the self similarity matrix in terms of pitches looks a lot more complex than that of Miss Alissa.

Clustering

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I thought it might be fun to see if I could cluster a day of songs into 6 clusters which resemble daily mixes. For this reason hierarchical clustering did not seem like the best option for it will be hard to pick 6 clusters with hierarchical clustering because it always branches out in 2. Meaning that I’d have to pick a depth with either 4 clusters or 8 clusters.

Therefore I have used k-means with k=6 to find 6 clusters.

I used energy, key, loudness, mode, speechiness, acousticness, instrumentalness, liveness, valence, tempo, time_signature, track.duration_ms and the MFCC’s as features for clustering.

Here you can see a graph of the energy and valence of the daily mixes (top) and the clusters (bottom). It is fairly apparent that valence and energy do not have a lot of say in distinguishing both daily mixes and k-means clusters.

Let’s find the important features.

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Training a classifier to determine cluster important features

Important features for clusters

Basic Idea

So now that we have an idea about what features we can extract and analyze using the spotify API, let’s return to the research questions:

1. Is there a way to predict my discover weekly playlist of the next week based on my daily mixes of the current week.

2. Is there a way to predict which daily mixes I listened to in the previous week(s) based on my discover weekly playlist.

Starting off, estimating my next discover weekly based on my daily mixes is going to be difficult. This will require me to compare the daily mixes to every song on spotify to find a fitting match for a discover weekly playlist. This is something spotify does very well and I doubt I will be able to recreate this wonderful algorithm for this project.

However, determining what daily mixes I listened to the most should be doable. Suppose we look at a discover weekly playlist and the daily mixes of the previous week. We can then write this down as an equation where each daily mix has a weight to determine how much I listened to it:

\(DW = DM_A^1 * W_1 + DM_B^1 * W_2 +...+ DM_F^7*W_{42}\)

Where each playlist is actually a set of features. Now an optimal solution where $ DW $ is estimated exactly likely does not exist. But this can be interpreted as a linear regression problem which we can solve.

Feature space

features on playlist level

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However, we are interested in the way playlist listen history and not track listen history. For simplicity we can take the mean of each feature for every playlist, resulting in the following feature space:

Least squares fitting

Calculated weights

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However, coming back to reality from our sidestep into linear algebra. Something to be said is that the use of mathematics is nice but it is important to keep it relevant. And negative values really do not make sense in this situation. A negative value would suggest that I intentionally ignored several playlists, which could be the case but I doubt spotify can measure this.

Non negative least squares

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Luckily I found a package online called “nnls” (non negative least squares) which calculates the least squares estimate using only positive values.

Unfortunately spotify’s recently played function does not let you see past the last 50 recently played songs so I do not know if these predictions are truly correct. But just looking at the weights do give me some confidence that they are not too bad. I usually listen to about three discover weeklies a day and sometimes I listen to all of them on a day but just a little bit of them all, or one daily mix more than the other. It looks like the weights do represent this.

Something that I do have to add and might be something to look into in the future is that the weights are not distributed equally among all days. As in: some days have a lot more weight to them than others. This could make sense, some days you listen to music less than others. However, all days should be treated equally.

Let’s take a look at how well the weights approximate the discover weekly mix.

What does it mean and how well did it do

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As you can see in the table, both least square methods perform pretty good when fitting for the discover weekly playlist.

The sum of squared error for the regular least square method is :

[1] 1.928051e-25

Whilst non negative least square gets:

[1] 5.553706e-29

This came to my suprise because the non negative least square method has more limitations (because it’s not allowed to be negative). This makes me think that my implementation of Least Squares may be flawed. But it also makes me happy that the more sensible approach with more realistic data performs better.